Estimating Price Elasticity Matrices

TL;DR

This paper introduces methods for estimating a structured elasticity matrix from demand and price data, using regularization and optimization techniques, with open-source Python code provided.

Contribution

It proposes and compares three optimization methods for estimating a low-rank plus diagonal elasticity matrix, enhancing demand modeling accuracy.

Findings

Gradient ascent method is faster due to efficient gradient computations.

All three methods yield similar likelihood results on data.

Synthetic data experiments show profit maximization aligns with likelihood maximization.

Abstract

The relationship between demand and prices of a set of products can be modeled as a linear mapping from logarithmic price changes to logarithmic changes in demand. We consider the problem of estimating the coefficient matrix of this mapping, the elasticity matrix, based on observed data consisting of real-valued prices and integer-valued demands. We regularize the estimation problem by imposing a factor model structure, i.e., that the elasticity matrix is diagonal plus low-rank, similar to factor models used for financial returns. Maximizing the likelihood of observations of this model is a bi-convex problem, meaning that there is a partition of the variables in which it is convex in each set when the other is fixed. We propose and compare three methods for finding a locally optimal estimate. The first is based on alternating maximization, and involves solving a sequence of convex problems. The second method exploits efficient gradient computations in a gradient ascent method. The final method is to use a general purpose nonlinear programming method. While all methods give the same result on numerical examples, the gradient ascent method is substantially faster, due to its efficient gradient evaluations. We report the likelihood with different hyper-parameters for synthetic and real-world data, with similar results. For synthetic data, we also report the realized profit when using the elasticity estimate for optimal pricing, which is maximized for the same set of hyper-parameters that also maximizes the likelihood. This paper is accompanied by easy to use open source Python code for fitting elasticity matrices to observed data, using our three numerical methods.